It is well-known that a real number \(\xi \)  being normal in base \(r\)  is equivalent to
the sequence \((r^k \xi )_k \)  being uniformly distributed mod 1
(cf. [1]}). The latter statement can be rephrased as \(\xi \bmod 1\)  is a generic point for the Lebesgue measure on the unit circle \(\mathbb {T}=\mathbb {R}/\mathbb {Z}\)  under the action of the map \(\mathbb {T}\rightarrow \mathbb {T}\)  given by \(x\mapsto rx\bmod 1\) .
Thanks to this dynamical point of view, in [2]} we were able to study the Borel complexity of sets of normal numbers in several numeration systems. We employed
a unified treatment for \(r\) -ary expansions, continued fraction expansions, \(\beta \) -expansions, and generalized GLS-expansions. In fact, we considered a dynamical system given by subshifts generated by all possible expansions of numbers in \([0,1]\)  in full-shifts over an at most countable set of digits (alphabet). In that setting normal numbers with respect to a given expansion method correspond to generic points of an appropriately selected shift-invariant measure on a subshift containing all possible expansions. It turns out that for these subshifts the set of generic points for any shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a \(\Pi ^0_3\) -complete set). The crucial dynamical feature we used was a feeble form of specification. All expansions named above generate subshifts with this specification property.
